Analyzing Motion in Free Fall: The Case of a Dropped Package from a Hot Air Balloon

The Physics Problem: Calculating Time and Impact Velocity

Question:

A hot air balloon is ascending at a rate of 12 meters per second and is at a height of 80 meters above the ground. At this point, a package is dropped from the balloon. The problem requires us to find:

1. The time it takes for the package to reach the ground.
2. The velocity of the package at the moment of impact with the ground.

Solution: Understanding Free Fall Dynamics

To solve this problem, we consider the motion of the package from the moment it is dropped:

Phase 1 - Calculating the Time to Reach the Ground

• The initial velocity of the package when dropped is 12 m/s (the same as the balloon).
• We take the ground level as y = 0 meters and the initial position of the package as y₀ = 80 meters.
• The time to reach the ground is calculated using the equation: y - y₀ = v₀t + (1/2)at², where 'a' is the acceleration due to gravity (-9.8 m/s²).

Phase 2 - Determining the Impact Velocity

• The velocity at the moment of impact is found using the equation: v = v₀ + at, where 'v₀' is the initial velocity, 'a' is the acceleration due to gravity, and 't' is the time calculated in Phase 1.

By applying these principles, we can find both the time taken for the package to reach the ground and its velocity upon impact.